After this lecture, you will be able to:

1. Understand and explain measures of central tendency (mean, median, mode) and spread (range, IQR, variance, standard deviation)
2. Calculate statistical measures using both R and Excel
3. Choose appropriate statistical measures for your data
4. Interpret statistical results in a biological context
5. Begin troubleshooting common programming issues

Getting Started Checklist

Essential Setup

• Installed R
• Installed RStudio
• Know some Markdown basics

From calculators to computers

Statistical softare in the 1970s

R in today’s world

• Leading tool in data science and statistics (although Python leads in majority of machine learning workflows)
• Over 22,000 packages on CRAN – extensive statistical capabilities
• Integration with other modern tools: Python, HTML, Javascript, Excel, AJAX…
• Meets modern academic standards of reproducibility and increasingly preferred by statisticians

RStudio IDE. Source: Januar Harianto

Why statistical programming?

Statistical programming combines statistics and computer code to:

1. Analyse data quickly and accurately – especially large datasets
2. Share methods and results with others
3. Automate complex calculations and visualise results clearly

It’s like having a powerful calculator that can help us tell stories about our data in a repeatable way.

Population vs Sample

Sampling in statistics

How well does a sample represent the population?

Some thoughts:

• Sample size: Larger samples are more likely to represent the population
• Sampling method: Random samples are more likely to be representative
• Population variability: More variability means larger samples are needed

Samples will vary

Different samples give different results – suppose we have a population of 1000 trees and we randomly sample 6 tree heights. If this is done 3 times, it is likely that the samples will be different:

set.seed(258) 
population <- rnorm(1000, mean = 20, sd = 5)

# create samples
sample1 <- sample(population, size = 6)
sample2 <- sample(population, size = 6)
sample3 <- sample(population, size = 6)

# show samples
for (i in 1:3) {
   cat(sprintf("Sample %d: ", i), get(paste0("sample", i)), "\n")
}

Sample 1:  21.66633 22.61768 22.79266 17.64633 14.50462 17.9679 
Sample 2:  15.60759 14.0909 17.89364 18.10461 20.48023 22.88689 
Sample 3:  17.75913 15.89302 26.43149 27.08996 14.99993 34.06894 

So how do we make sense of these samples?

Descriptive statistics

We can describe our samples using:

1. Measures of central tendency – describe the “typical” value
   • mean, median, mode
2. Measures of spread – describe how much the data varies
   • standard deviation, variance (commonly used)
   • range, quartiles, IQR (for unique cases)
3. Measures of uncertainty – describe how confident we are in our estimates
   • standard error
   • confidence intervals

Mean – also known as the average

The mean is what most people call the “average”:

• Add up all your numbers
• Divide by how many numbers you have

Mean in R

We can save a group of numbers in a vector called scores in R:

# Our test scores
scores <- c(80, 85, 90, 95)

# manual calculation
(80 + 85 + 90 + 95) / 4

[1] 87.5

# Alternative way
sum(scores) / length(scores)

[1] 87.5

mean(scores)

[1] 87.5

Mean in Excel

Median – the middle value

The median is the middle number when your data is in order:

1. First, put your numbers in order
2. Find the middle value
3. If you have an even number of values, take the average of the two middle numbers

Median in R

R does all the ordering and finding the middle for us:

# House prices
prices <- c(450, 1100, 480, 460, 470, 420, 1400, 450, 470)

# Find median
median(prices)

[1] 470

# Compare with mean
mean(prices)

[1] 633.3333

Median in Excel

Excel provides two main ways to find the median:

1. Using MEDIAN function

   =MEDIAN(A1:A9)

   • Type =MEDIAN(
   • Select your data range
   • Press Enter

2. Alternative method

   • Sort your data first (use the Sort functionality in the Data tab)
   • Find middle value(s)
   • If even number of values, average the middle two

Mode – most frequent value

Calculating the mode in R

There is no built-in function to calculate the mode, so we use the modeest package:

if(!require("modeest")) install.packages("modeest")

Loading required package: modeest

library(modeest)

df <- c(1, 2, 3, 3, 4, 5, 5, 5, 6)
mlv(df, method = "mfv")  # most frequent value

[1] 5

freq_table <- table(df) # Count frequencies of each value
# Find which value(s) appear most often
modes <- as.numeric(names(freq_table[freq_table == max(freq_table)]))
modes

[1] 5

sorted_df <- sort(df) # Sort the vector first
runs <- rle(sorted_df) # Use run-length encoding to find sequences
modes <- runs$values[runs$lengths == max(runs$lengths)] # Find the value(s) with max length
modes

[1] 5

unique_vals <- factor(df) # Create a factor of unique values
counts <- tapply(df, unique_vals, length) # Count occurrences using tapply
modes <- as.numeric(names(counts[counts == max(counts)])) # Find which values have the maximum count
modes

[1] 5

Mode in Excel

Excel provides several methods to find the mode:

1. MODE.SNGL function (single mode)

=MODE.SNGL(A1:A10)

• Returns most frequent value
• Returns #N/A if no repeats

2. MODE.MULT function (multiple modes)

=MODE.MULT(A1:A10)

• Returns array of modes
• Press Ctrl+Shift+Enter

A biological example

Source: Adobe Stock # 85659279

Imagine sampling seagrass blade lengths from two different sites in a marine ecosystem, and they have the same mean length of 15.2 cm. Are both sites the same?

Comparing Different Measures

# Plot seagrass lengths
library(ggplot2)
library(patchwork)

Warning: package 'patchwork' was built under R version 4.4.1

seagrass_protected <- c(15.2, 15.0, 15.3, 15.1, 15.2)
seagrass_exposed <- c(12.0, 18.0, 14.5, 16.5, 15.0)

# Create plots for both sites
p1 <- ggplot() +
   geom_point(aes(x = 1:5, y = seagrass_protected), size = 3) +
   geom_hline(yintercept = mean(seagrass_protected), linetype = "dashed", color = "red") +
   labs(title = "Site A: Protected Bay", x = "Measurement", y = "Length (cm)") +
   ylim(10, 20)

p2 <- ggplot() +
   geom_point(aes(x = 1:5, y = seagrass_exposed), size = 3) +
   geom_hline(yintercept = mean(seagrass_exposed), linetype = "dashed", color = "red") +
   labs(title = "Site B: Wave-exposed Coast", x = "Measurement", y = "Length (cm)") +
   ylim(10, 20)

# Combine plots side by side
p1 + p2

Why do we need measures of spread?

• Central tendency (mean, median, mode) only tells part of the story
• Spread tells us how much variation exists in our data
• Different measures of spread tell us different things:
  • Range: Overall spread of data
  • IQR: Spread of middle 50% of data
  • Variance: Average squared deviation from mean
  • Standard deviation: Average deviation in original units

Range – The simplest measure of spread

# Create our seagrass data
seagrass_protected <- c(15.2, 15.0, 15.3, 15.1, 15.2)  # Protected bay
seagrass_exposed <- c(12.0, 18.0, 14.5, 16.5, 15.0)    # Wave-exposed coast

# Calculate ranges
cat("Protected bay range:", diff(range(seagrass_protected)), "cm\n")

Protected bay range: 0.3 cm

cat("Wave-exposed range:", diff(range(seagrass_exposed)), "cm\n")

Wave-exposed range: 6 cm

Interquartile range (IQR): The middle 50%

The IQR tells us how spread out the middle 50% of our data is:

# Get quartiles for protected bay
quantile(seagrass_protected)

  0%  25%  50%  75% 100% 
15.0 15.1 15.2 15.2 15.3 

• 25% of data below Q1 (1st quartile)
• 75% of data below Q3 (3rd quartile)
• IQR = Q3 - Q1

Why use IQR?

• Ignores extreme values
• Works with skewed data
• More stable than range

Comparing Sites Using IQR

# Compare IQRs
pbay <- IQR(seagrass_protected)
pbay

[1] 0.1

exbay <- IQR(seagrass_exposed)
exbay

[1] 2

• Protected bay IQR: 0.1 cm
• Wave-exposed IQR: 2 cm

Variance: a detailed measure of spread

Variance measures how far data points are spread from their mean by:

1. Finding how far each point is from the mean
2. Squaring these distances (to handle negative values)
3. Taking the average of these squared distances

Why use variance?

• Uses all data points (unlike IQR)
• Less sensitive to outliers than range
• Shows total spread in both directions

Calculating Variance in R

# Calculate variance for both sites
cat("Protected bay variance:", var(seagrass_protected), "cm²\n")

Protected bay variance: 0.013 cm²

cat("Wave-exposed variance:", var(seagrass_exposed), "cm²\n")

Wave-exposed variance: 5.075 cm²

Standard deviation: a more interpretable measure

Standard deviation (SD, or \sigma for population, s for sample) is the square root of variance:

• Tells us the “typical distance” from the mean
• Easy to understand - similar to saying “± value” after a mean
• Small SD means values cluster closely around mean
• Large SD means values are more spread out

When and Why to Use It

• Values are in the same units as your data (unlike variance)
• Perfect for describing natural variation (height, weight, temperature)
• Used in many statistical tests
• Great for comparing different groups or datasets

Interpreting standard deviation (with R)

We can describe our seagrass lengths using mean ± standard deviation:

# Protected bay
mean_p <- mean(seagrass_protected)
sd_p <- sd(seagrass_protected)
cat("Protected bay:", round(mean_p, 1), "±", round(sd_p, 2), "cm\n")

Protected bay: 15.2 ± 0.11 cm

# Wave-exposed
mean_e <- mean(seagrass_exposed)
sd_e <- sd(seagrass_exposed)
cat("Wave-exposed:", round(mean_e, 1), "±", round(sd_e, 2), "cm\n")

Wave-exposed: 15.2 ± 2.25 cm

Comparing Spread Measures

Key Observations

• Wave-exposed site shows consistently more variation
• Each measure gives a different perspective
• Choose based on your data and goals
• Standard deviation is most commonly used in research papers

Basic Measures of Spread in Excel

Common Excel functions for measuring spread:

1. Range: Use MAX() and MIN()

=MAX(A1:A10) - MIN(A1:A10)

2. Quartiles and IQR: Use QUARTILE.INC()

For Q1: =QUARTILE.INC(A1:A10, 1)
For Q3: =QUARTILE.INC(A1:A10, 3)
For IQR: =QUARTILE.INC(A1:A10, 3) - QUARTILE.INC(A1:A10, 1)

Advanced Measures of Spread in Excel

Statistical functions for variance and standard deviation:

1. Sample Variance: Use VAR.S()

=VAR.S(A1:A10)

2. Sample Standard Deviation: Use STDEV.S()

=STDEV.S(A1:A10)

Core Reading

• Quinn & Keough (2024). Experimental Design and Data Analysis for Biologists. Cambridge University Press. Chapter 2: Things to know before proceeding.
• Canvas site for lecture notes and additional resources